A piece of wire of resistance R is cut into 6 equal parts. These parts are then connected in parallel. If the equivalent resistance of this combination is RI , then find the ratio of R and RI .
Answers
Answer:
Step 1: Resistance of each part after cutting
We know that, Resistance R=ρ
A
ℓ
Since, R is directly proportional to length of the wire.
Therefore the resistance of every part after cutting the wire in 5 parts is
R
1
=R
2
=R
3
=R
4
=R
5
=
5
R
Step 2: Equivalent Resistance
When all the resistances are connected in parallel, let R' be the equivalent resistance
So,
R
′
1
=
R
1
1
+
R
2
1
+
R
3
1
+
R
4
1
+
R
5
1
⇒
R
′
1
=
R
5
+
R
5
+
R
5
+
R
5
+
R
5
⇒R
′
=
25
R
⇒
R
′
R
=25
Answer:
Let’s say the total resistance of the wire before cutting wasR.
When we cut the wire into 6 parts, each one has a length of L/6 and all other parameters remain unchanged.
So the resistance of each new segment isR/6.
Connecting them parallel gives a network as shown in figure attached.
Now since the question asks for The effective resistance of this combination, lets recollect that the effective resistance of a parallel combination of n resistances - R1,R2,R3, . . . Rn is given by: 1/Reff=1/R1+1/R2+...+1/Rn
So if R′ is the resistance of this new combination, then:
1/R′=1/(R/6)+1/(R/6)+1/(R/6)+1/(R/6)+1/(R/6)+1/(R/6)
Simplifying this expression a bit gives us:
1R′=6/(R/6)
R′=R/36
So the ratio R/R′=36.